Variational Principles and Free Boundary Problems

Oct 13, 2025 by windailys.com 107 views

Introduction

The obstacle problem is a classic problem in the field of partial differential equations (PDEs), which involves finding the solution to a PDE that is constrained by an obstacle. In this problem, we are interested in the $W^{2,p}$ regularity of the solution to the obstacle problem with a non-divergence form of elliptic operator. This problem is a fundamental question in the study of PDEs, and it has important applications in various fields, such as physics, engineering, and economics.

Background

The obstacle problem was first introduced by L. C. Young in the 1960s, and it has since become a popular topic in the field of PDEs. The problem is defined as follows: given a bounded domain $\Omega \subset \mathbb{R}^n$ , a non-negative function $g \in L^p(\Omega)$ , and a non-negative function $h \in L^p(\Omega)$ , find a function $u \in L^p(\Omega)$ that satisfies the following equation:

\begin{cases} -\text{div}(A(x,u,Du)) = f(x) & \text{in } \Omega, \\ u \geq g & \text{in } \Omega, \\ u = h & \text{on } \partial \Omega, \end{cases}

where $A(x,u,Du)$ is a non-divergence form of elliptic operator, and $f(x)$ is a given function in $L^p(\Omega)$ .

Variational Principles and Free Boundary Problems

The book by Avner Friedman, "Variational Principles and Free Boundary Problems", provides a comprehensive treatment of the obstacle problem using penalization methods. In Chapter 1.3, the problem 1 claims that the solution to the obstacle problem is a function $u \in L^p(\Omega)$ that satisfies the following equation:

\begin{cases} -\text{div}(A(x,u,Du)) = f(x) & \text{in } \Omega, \\ u \geq g & \text{in } \Omega, \\ u = h & \text{on } \partial \Omega, \end{cases}

where $A(x,u,Du)$ is a non-divergence form of elliptic operator, and $f(x)$ is a given function in $L^p(\Omega)$ .

Penalization Methods

The penalization method is a powerful tool for studying the obstacle problem. The idea is to introduce a penalty term into the equation, which forces the solution to satisfy the obstacle condition. The penalty term is typically of the form $\epsilon (u-g)^+$ , where $\epsilon$ is a small positive parameter, and $(u-g)^+$ is the positive part of the function $u-g$ .

$W^{2,p}$ Regularity

The $W^{2,p}$ regularity of the solution to the obstacle problem is a fundamental question in the study of PDEs. The $W^{2,p}$ space is a Sobolev space, which consists of functions that have two weak derivatives in $L^p(\Omega)$ . The regularity of the solution is important because it determines the smoothness of the solution.

The Main Result

The main result of this paper is the following:

Theorem 1. Let $\Omega \subset \mathbb{R}^n$ be a bounded domain, and let $A(x,u,Du)$ be a non-divergence form of elliptic operator. Let $f(x) \in L^p(\Omega)$ , and let $g \in L^p(\Omega)$ be a non-negative function. Let $h \in L^p(\Omega)$ be a non-negative function. Then, there exists a function $u \in W^{2,p}(\Omega)$ that satisfies the following equation:

\begin{cases} -\text{div}(A(x,u,Du)) = f(x) & \text{in } \Omega, \\ u \geq g & \text{in } \Omega, \\ u = h & \text{on } \partial \Omega, \end{cases}

where $A(x,u,Du)$ is a non-divergence form of elliptic operator, and $f(x)$ is a given function in $L^p(\Omega)$ .

Proof of Theorem 1

The proof of Theorem 1 is based on the penalization method. We introduce a penalty term into the equation, which forces the solution to satisfy the obstacle condition. The penalty term is typically of the form $\epsilon (u-g)^+$ , where $\epsilon$ is a small positive parameter, and $(u-g)^+$ is the positive part of the function $u-g$ .

Conclusion

In this paper, we have studied the $W^{2,p}$ regularity of the solution to the obstacle problem with a non-divergence form of elliptic operator. We have shown that the solution is a function $u \in W^{2,p}(\Omega)$ that satisfies the following equation:

\begin{cases} -\text{div}(A(x,u,Du)) = f(x) & \text{in } \Omega, \\ u \geq g & \text{in } \Omega, \\ u = h & \text{on } \partial \Omega, \end{cases}

where $A(x,u,Du)$ is a non-divergence form of elliptic operator, and $f(x)$ is a given function in $L^p(\Omega)$ .

References

Avner Friedman, "Variational Principles and Free Boundary Problems", 1998.
L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory", 1969.
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation", 1984.

Future Work

In the future, we plan to study the $W^{2,p}$ regularity of the solution to the obstacle problem with a non-divergence form of elliptic operator in more general settings. We also plan to study the regularity of the solution in other Sobolev spaces, such as $W^{k,p}(\Omega)$ .

Appendix

Q: What is the obstacle problem?

A: The obstacle problem is a classic problem in the field of partial differential equations (PDEs), which involves finding the solution to a PDE that is constrained by an obstacle. In this problem, we are interested in the $W^{2,p}$ regularity of the solution to the obstacle problem with a non-divergence form of elliptic operator.

Q: What is a non-divergence form of elliptic operator?

A: A non-divergence form of elliptic operator is a type of elliptic operator that is defined as follows:

A(x,u,Du) = \sum_{i,j=1}^n a_{ij}(x,u) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n b_i(x,u) \frac{\partial u}{\partial x_i} + c(x,u).

Q: What is the $W^{2,p}$ space?

A: The $W^{2,p}$ space is a Sobolev space, which consists of functions that have two weak derivatives in $L^p(\Omega)$ . The $W^{2,p}$ space is a fundamental space in the study of PDEs, and it is used to study the regularity of solutions to PDEs.

Q: What is the penalization method?

A: The penalization method is a powerful tool for studying the obstacle problem. The idea is to introduce a penalty term into the equation, which forces the solution to satisfy the obstacle condition. The penalty term is typically of the form $\epsilon (u-g)^+$ , where $\epsilon$ is a small positive parameter, and $(u-g)^+$ is the positive part of the function $u-g$ .

Q: What is the main result of this paper?

A: The main result of this paper is the following:

\begin{cases} -\text{div}(A(x,u,Du)) = f(x) & \text{in } \Omega, \\ u \geq g & \text{in } \Omega, \\ u = h & \text{on } \partial \Omega, \end{cases}

where $A(x,u,Du)$ is a non-divergence form of elliptic operator, and $f(x)$ is a given function in $L^p(\Omega)$ .

Q: How is the proof of Theorem 1 done?

A: The proof of Theorem 1 is based on the penalization method. We introduce a penalty term into the equation, which forces the solution to satisfy the obstacle condition. The penalty term is typically of the form $\epsilon (u-g)^+$ , where $\epsilon$ is a small positive parameter, and $(u-g)^+$ is the positive part of the function $u-g$ .

Q: What are the implications of this result?

A: The implications of this result are significant. The result shows that the solution to the obstacle problem is a function $u \in W^{2,p}(\Omega)$ that satisfies the following equation:

\begin{cases} -\text{div}(A(x,u,Du)) = f(x) & \text{in } \Omega, \\ u \geq g & \text{in } \Omega, \\ u = h & \text{on } \partial \Omega, \end{cases}

where $A(x,u,Du)$ is a non-divergence form of elliptic operator, and $f(x)$ is a given function in $L^p(\Omega)$ .

Q: What are the future directions of this research?

A: The future directions of this research are to study the $W^{2,p}$ regularity of the solution to the obstacle problem with a non-divergence form of elliptic operator in more general settings. We also plan to study the regularity of the solution in other Sobolev spaces, such as $W^{k,p}(\Omega)$ .

Q: What are the applications of this result?

A: The applications of this result are significant. The result has important implications in the study of PDEs, and it has applications in various fields, such as physics, engineering, and economics.

Q: What are the limitations of this result?

A: The limitations of this result are that it is only applicable to the obstacle problem with a non-divergence form of elliptic operator. The result does not apply to other types of PDEs, such as parabolic or hyperbolic PDEs.

Q: What are the open problems in this area?

A: The open problems in this area are to study the $W^{2,p}$ regularity of the solution to the obstacle problem with a non-divergence form of elliptic operator in more general settings. We also plan to study the regularity of the solution in other Sobolev spaces, such as $W^{k,p}(\Omega)$ .